You are making a fascinating mistake, and I may make a separate post about it, even though it’s not particularly related to anthropics and just a curious detail about probability theory, which in retrospect I relize I was confused myself about. I’d recommend you to meditate on it for a while. You already have all the information required to figure it out. You just need to switch yourself from the “argument mode” to “investigation mode”.
Here are a couple more hints that you may find useful.
1) Suppose you observed number 71 on a random number generator that produces numbers from 0 to 99.
Is it
1 in 100 occurence because the number is exactly 71?
1 in 50 occurence becaue the number consist of these two digits: 7 and 1?
1 in 10 occurence because the first digit is 7?
1 in 2 occurence because the number is more or equal 50?
1 in n occurence because it’s possible to come with some other arbitrary rule?
What determine which case is actually true?
2) Suppose you observed a list of numbers with length n, produced by this random number generator. The probability that exactly this series is produced is 1/100n
At what n are you completely shocked and in total disbelief about your reality, after all you’ve just observed an event that your model of reality claims to be extremely improbable?
Would you be more shocked if all the numbers in this list are the same? If so why?
Can you now produce arbitrary improbable events just by having a random number generator? In what sense are these events have probability 1/100n if you can witness as many of them as you want any time?
You do not need to tell me the answers. It’s just something I believe will be helpful for you to honestly think about.
To find the posterior probability of Heads, given what you have observed, you combine the prior probability with the likelihood for Heads vs. Tails based on everything that you have observed.
Here is the last hint, actually I have a feeling that this just spoils the solution outright so it’s in rot13:
Gur bofreingvbaf “Enaqbz ahzore trarengbe cebqhprq n ahzore” naq “Enaqbz ahzore trarengbe cebqhprq gur rknpg ahzore V’ir thrffrq” ner qvssrerag bofreingvbaf. Lbh pna bofreir gur ynggre bayl vs lbh’ir thrffrq n ahzore orsberunaq. Lbh znl guvax nobhg nf nal bgure novyvgl gb rkgenpg vasbezngvba sebz lbhe raivebazrag.
Fhccbfr jura gur pbva vf Gnvyf gur ebbz unf terra jnyyf naq jura vg’f Urnqf gur ebbz unf oyhr jnyyf. N crefba jub xabjf nobhg guvf naq vfa’g pbybe oyvaq pna thrff gur erfhyg bs n pbva gbff cresrpgyl. N pbybe oyvaq crefba jub xabjf nobhg guvf ehyr—pna’g. Rira vs gurl xabj gung gur ebbz unf fbzr pbybe, gurl ner hanoyr gb rkrphgr gur fgengrtl “thrff Gnvyf rirel gvzr gur ebbz vf terra”.
N crefba jub qvqa’g thrff n ahzore orsberunaq qbrfa’g cbffrff gur novyvgl gb bofreir rirag “Enaqbz ahzore trarengbe cebqhprq gur rknpg ahzore V’ir thrffrq” whfg nf n pbybe oyvaq crefba qbrfa’g unir na novyvgl gb bofreir na rirag “Gur ebbz vf terra”.
I think if Beauty isn’t a conscious being, it doesn’t make much sense to talk about how she should reason regarding philosophical arguments about probability.
The Beauty doesn’t need to experience qualia or be self aware to have meaningful probability estimate.
I’ll just mention that probability is supposed to be useful. And if you extend the problem to allow Beauty to make bets, in various scenarios, the bets the make Beauty the most money are the ones she will make by assessing the probability of Heads to be 1⁄3 and then applying standard decision theory.
Betting arguments are not particularly helpful. They are describing the motte—specific scoring rule, and not the actual ability to guess the outcome of the coin toss in the experiment. As I’ve written in the post itself:
As long as we do not claim that this fact gives an ability to predict the result of the coin toss better than chance, then we are just using different definitions, while agreeing on everything. We can translate from Thirder language to mine and back without any problem. Whatever betting schema is proposed, all other things being equal, we will agree to the same bets.
That is, if betting happens every day, Halfers and Double Halfers need to weight the odds by the number of bets, while Thirders already include this weighting in their definitions “probability”. On the other hand, if only one bet per experiment counts, suddenly it’s thirders who need to discount this weighting from their “probability” and Halfers and Double Halfers who are fine by default.
There are rules for how to do arithmetic. If you want to get the right answer, you have to follow them. So, when adding 18 and 17, you can’t just decide that you don’t like to carry 1s today, and hence compute that 18+17=25.
Similarly, there are rules for how to do Bayesian probability calculations. If you want to get the right answer, you have to follow them. One of the rules is that the posterior probability of something is found by conditioning on all the data you have. If you do a clinical trial with 1000 subjects, you can’t just decide that you’d like to compute the posterior probability that the treatment works by conditioning on the data for just the first 700.
If you’ve seen the output of a random number generator, and are using this to compute a posterior probability, you condition on the actual number observed, say 71. You do not condition on any of the other events you mention, because they are less informative than the actual number—conditioning on them would amount to ignoring part of the data. (In some circumstances, the result of conditioning on all the data may be the same as the result of conditioning on some function of the data—when that function is a “sufficient statistic”, but it’s always correct to condition on all the data.)
This is absolutely standard Bayesian procedure. There is nothing in the least bit controversial about it. (That is, it is definitely how Bayesian inference works—there are of course some people who don’t accept that Bayesian inference is the right thing to do.)
Similarly, there are certain rules for how to apply decision theory to choose an action to maximize your expected utility, based on probability judgements that you’ve made.
If you compute probabilities incorrectly, and then incorrectly apply decision theory to choose an action based on these incorrect probabilities, it is possible that your two errors will cancel out. That is actually rather likely if you have other ways of telling what the right answer is, and hence have the opportunity to make ad hoc (incorrect) alterations to how you apply decision theory in order to get the right decision with the wrong probabilities.
If you’d like to outline some specific betting scenario for Sleeping Beauty, I’ll show you how applying decision theory correctly produces the right action only if Beauty judges the probability of Heads to be 1⁄3.
You are making a fascinating mistake, and I may make a separate post about it, even though it’s not particularly related to anthropics and just a curious detail about probability theory, which in retrospect I relize I was confused myself about. I’d recommend you to meditate on it for a while. You already have all the information required to figure it out. You just need to switch yourself from the “argument mode” to “investigation mode”.
Here are a couple more hints that you may find useful.
1) Suppose you observed number 71 on a random number generator that produces numbers from 0 to 99.
Is it
1 in 100 occurence because the number is exactly 71?
1 in 50 occurence becaue the number consist of these two digits: 7 and 1?
1 in 10 occurence because the first digit is 7?
1 in 2 occurence because the number is more or equal 50?
1 in n occurence because it’s possible to come with some other arbitrary rule?
What determine which case is actually true?
2) Suppose you observed a list of numbers with length n, produced by this random number generator. The probability that exactly this series is produced is 1/100n
At what n are you completely shocked and in total disbelief about your reality, after all you’ve just observed an event that your model of reality claims to be extremely improbable?
Would you be more shocked if all the numbers in this list are the same? If so why?
Can you now produce arbitrary improbable events just by having a random number generator? In what sense are these events have probability 1/100n if you can witness as many of them as you want any time?
You do not need to tell me the answers. It’s just something I believe will be helpful for you to honestly think about.
Here is the last hint, actually I have a feeling that this just spoils the solution outright so it’s in rot13:
Gur bofreingvbaf “Enaqbz ahzore trarengbe cebqhprq n ahzore” naq “Enaqbz ahzore trarengbe cebqhprq gur rknpg ahzore V’ir thrffrq” ner qvssrerag bofreingvbaf. Lbh pna bofreir gur ynggre bayl vs lbh’ir thrffrq n ahzore orsberunaq. Lbh znl guvax nobhg nf nal bgure novyvgl gb rkgenpg vasbezngvba sebz lbhe raivebazrag.
Fhccbfr jura gur pbva vf Gnvyf gur ebbz unf terra jnyyf naq jura vg’f Urnqf gur ebbz unf oyhr jnyyf. N crefba jub xabjf nobhg guvf naq vfa’g pbybe oyvaq pna thrff gur erfhyg bs n pbva gbff cresrpgyl. N pbybe oyvaq crefba jub xabjf nobhg guvf ehyr—pna’g. Rira vs gurl xabj gung gur ebbz unf fbzr pbybe, gurl ner hanoyr gb rkrphgr gur fgengrtl “thrff Gnvyf rirel gvzr gur ebbz vf terra”.
N crefba jub qvqa’g thrff n ahzore orsberunaq qbrfa’g cbffrff gur novyvgl gb bofreir rirag “Enaqbz ahzore trarengbe cebqhprq gur rknpg ahzore V’ir thrffrq” whfg nf n pbybe oyvaq crefba qbrfa’g unir na novyvgl gb bofreir na rirag “Gur ebbz vf terra”.
The Beauty doesn’t need to experience qualia or be self aware to have meaningful probability estimate.
Betting arguments are not particularly helpful. They are describing the motte—specific scoring rule, and not the actual ability to guess the outcome of the coin toss in the experiment. As I’ve written in the post itself:
That is, if betting happens every day, Halfers and Double Halfers need to weight the odds by the number of bets, while Thirders already include this weighting in their definitions “probability”. On the other hand, if only one bet per experiment counts, suddenly it’s thirders who need to discount this weighting from their “probability” and Halfers and Double Halfers who are fine by default.
There are rules for how to do arithmetic. If you want to get the right answer, you have to follow them. So, when adding 18 and 17, you can’t just decide that you don’t like to carry 1s today, and hence compute that 18+17=25.
Similarly, there are rules for how to do Bayesian probability calculations. If you want to get the right answer, you have to follow them. One of the rules is that the posterior probability of something is found by conditioning on all the data you have. If you do a clinical trial with 1000 subjects, you can’t just decide that you’d like to compute the posterior probability that the treatment works by conditioning on the data for just the first 700.
If you’ve seen the output of a random number generator, and are using this to compute a posterior probability, you condition on the actual number observed, say 71. You do not condition on any of the other events you mention, because they are less informative than the actual number—conditioning on them would amount to ignoring part of the data. (In some circumstances, the result of conditioning on all the data may be the same as the result of conditioning on some function of the data—when that function is a “sufficient statistic”, but it’s always correct to condition on all the data.)
This is absolutely standard Bayesian procedure. There is nothing in the least bit controversial about it. (That is, it is definitely how Bayesian inference works—there are of course some people who don’t accept that Bayesian inference is the right thing to do.)
Similarly, there are certain rules for how to apply decision theory to choose an action to maximize your expected utility, based on probability judgements that you’ve made.
If you compute probabilities incorrectly, and then incorrectly apply decision theory to choose an action based on these incorrect probabilities, it is possible that your two errors will cancel out. That is actually rather likely if you have other ways of telling what the right answer is, and hence have the opportunity to make ad hoc (incorrect) alterations to how you apply decision theory in order to get the right decision with the wrong probabilities.
If you’d like to outline some specific betting scenario for Sleeping Beauty, I’ll show you how applying decision theory correctly produces the right action only if Beauty judges the probability of Heads to be 1⁄3.